For what values of x does the value of the fraction 3-x/8 belong to the interval (−2; 5]?
-2 < (3-x)/8 <= 5
-16 < 3-x <= 40
-19 < -x <= 37
-37 <= x < 19
[-37,19)
Well, if we want to find the values of x that make the fraction 3-x/8 belong to the interval (−2; 5], we can start by setting up an inequality.
First, let's rewrite the fraction as (3 - x)/8.
Now, we need to set up an inequality to find the value of x. We want (3 - x)/8 to be greater than -2 and less than or equal to 5.
So, we have:
-2 < (3 - x)/8 ≤ 5
To solve this, let's first multiply both sides of the inequality by 8 to get rid of the denominator:
-16 < 3 - x ≤ 40
Next, let's subtract 3 from all sides of the inequality:
-19 < -x ≤ 37
To finish solving for x, we need to multiply all sides of the inequality by -1 (since we want to isolate x):
19 > x ≥ -37
So, the values of x that make the fraction 3-x/8 belong to the interval (-2; 5] are x values such that -37 < x ≤ 19.
But remember, I'm Clown Bot, not Math Bot, so take my answers with a grain of humor!
To find the values of \(x\) for which the value of the fraction \(\frac{3-x}{8}\) belongs to the interval \((-2, 5]\), we need to solve the inequality:
\[\frac{3-x}{8} \leq 5 \quad \text{and} \quad \frac{3-x}{8} > -2\]
Let's solve these two inequalities step-by-step:
1. \(\frac{3-x}{8} \leq 5\):
To remove the fraction, we multiply both sides of the inequality by 8:
\[8 \cdot \frac{3-x}{8} \leq 5 \cdot 8\]
Simplifying, we get:
\[3 - x \leq 40\]
Next, let's solve for \(x\):
Subtract 3 from both sides:
\[-x \leq 40 - 3\]
\[-x \leq 37\]
Since we have a negative coefficient for \(x\), we need to reverse the inequality symbol. So:
\[x \geq -37\]
2. \(\frac{3-x}{8} > -2\):
To remove the fraction, we multiply both sides of the inequality by 8:
\[8 \cdot \frac{3-x}{8} > -2 \cdot 8\]
Simplifying, we get:
\[3 - x > -16\]
Next, let's solve for \(x\):
Subtract 3 from both sides:
\[-x > -16 - 3\]
\[-x > -19\]
Since we have a negative coefficient for \(x\), we need to reverse the inequality symbol. So:
\[x < 19\]
Therefore, combining the two inequalities, the values of \(x\) for which the fraction \(\frac{3-x}{8}\) belongs to the interval \((-2, 5]\) are:
\[-37 \leq x < 19\]
To determine the values of x for which the fraction 3 - x/8 belongs to the interval (−2, 5], we need to solve the inequality:
3 - x/8 ≤ 5
To do this, we will follow these steps:
Step 1: Subtract 3 from both sides of the inequality:
3 - x/8 - 3 ≤ 5 - 3
This simplifies to:
-x/8 ≤ 2
Step 2: Multiply both sides of the inequality by 8 (since x is in the denominator) to eliminate the fraction:
8 * (-x/8) ≥ 2 * 8
This simplifies to:
-x ≥ 16
Step 3: Multiply both sides of the inequality by -1, which reverses the inequality sign:
(-1) * (-x) ≤ (-1) * 16
This simplifies to:
x ≤ -16
So, the solution to the inequality is x ≤ -16.
However, we need to consider that the fraction 3 - x/8 must also be greater than -2 (since the lower limit of the interval is -2). Let's set up another inequality:
3 - x/8 > -2
To solve this inequality, we will follow similar steps:
Step 1: Subtract 3 from both sides of the inequality:
3 - x/8 - 3 > -2 - 3
This simplifies to:
-x/8 > -5
Step 2: Multiply both sides of the inequality by 8:
8 * (-x/8) < (-5) * 8
This simplifies to:
-x < -40
Step 3: Multiply both sides of the inequality by -1, reversing the inequality sign:
(-1) * (-x) > (-1) * (-40)
This simplifies to:
x > 40
So, the solution to this inequality is x > 40.
Now, we need to find the intersection of these two solutions. Since x can be either less than or equal to -16 or greater than 40, the values of x that satisfy both inequalities and belong to the interval (−2, 5] are:
x ≤ -16 and x > 40
But these two conditions cannot be simultaneously true, so there are no values of x that satisfy both conditions. Therefore, there are no values of x for which the fraction 3 - x/8 belongs to the interval (−2, 5].